Chapter 4.6, Problem 51WE

To determine

To Prove:

To show that $x^2+\left(k+1\right)x+k$ is reducible for all positive integers k.

Explanation of Solution

Given information:

The given polynomials is $x^2+\left(k+1\right)x+k$ .

Consider the polynomial $x^2+\left(k+1\right)x+k$ ,

In order to factor a polynomial, we need 2 numbers that add up to the co-efficient of the linear term and multiply to equal the product of the co-efficient of the quadratic term and constant.

For instance, consider the polynomial $x^2+4x+3$ , two numbers that add up to 4 and multiply to 3 are 3 and 1. Therefore, this can be factored to $\left(x+3\right)\left(x+1\right)$ .

Therefore, we can conclude for all the cases that $x^2+\left(k+1\right)x=k$ is reducible for all positive k since it can be simplified to $\left(x+k\right)\left(x+1\right)$ .

Hence, the polynomial $x^2+\left(k+1\right)x+k$ is reducible for all positive integers k.